Locally Lipschitz BSDE with jumps and related Kolmogorov equation

We study a backward SDE driven by a jump Markov process (BSDEJ for short) whose generator may be locally Lipschitz or of logarithmic growth in (y, z(.))-variables. The existence, uniqueness and stability theorems to such BSDEJs are established. We essentially approximate the initial problem by constructing a suitable sequence of BSDEJs with globally Lipschitz generators for which the existence and uniqueness of solutions hold. By passing to the limits, we show the existence and uniqueness of solutions to the original problems. We apply our main results to prove the existence of a unique solution to the Kolmogorov equation of the Markov process.

Automated summary

We study backward stochastic differential equations driven by jump Markov processes and establish the theorems of existence, uniqueness, and stability. By constructing a sequence of BSDEJs with globally Lipschitz generators, we approximate the initial problem and prove the existence and uniqueness of solutions. Furthermore, we apply our main results to prove the existence of a unique solution to the Kolmogorov equation of the Markov process.